Given that 6,9,131/2,201/4 form a sequence of numbers. Find the sum of its 13 terms
I don't know the answer so help me
6,9,131/2,201/4
the way you typed it, I see no pattern but ..
6,9,13 1/2,20 1/4, ...
= 6, 9, 27/2, 81/2, ..
a GP with a = 6, r = 3/2
sum(13) = a(r^13 - 1)/(r-1)
= 6( (3/2)^13 - 1)/(3/2 - 1)
= appr 2323.434
To find the sum of the sequence, we need to first identify the sequence and then apply the formula for the sum of an arithmetic sequence.
Looking at the given terms:
6, 9, 131/2, 201/4
We can see that each term is obtained by dividing the previous term by 2 and then adding a certain value. Let's denote the first term as a and the difference between consecutive terms as d.
a = 6 (first term)
d = 9 - 6 = 3 (difference between the first and second term)
Now, we can calculate the 13th term using the formula:
an = a + (n - 1) * d
a13 = 6 + (13 - 1) * 3
= 6 + 12 * 3
= 6 + 36
= 42
The 13th term of the sequence is 42.
Next, we can use the formula for the sum of an arithmetic sequence to find the sum of the 13 terms:
Sn = (n / 2) * (2a + (n - 1) * d)
S13 = (13 / 2) * (2 * 6 + (13 - 1) * 3)
= (13 / 2) * (12 + 12 * 3)
= (13 / 2) * (12 + 36)
= (13 / 2) * 48
= 13 * 24
= 312
Therefore, the sum of the 13 terms in the sequence is 312.
To find the sum of the 13 terms in the given sequence, we need to first identify the pattern in the sequence.
The given sequence is: 6, 9, 131/2, 201/4
The terms in the sequence seem to be increasing, but not in a consistent manner. However, if we look closely, we can see that the pattern is alternating between adding 3 and then adding 2. Let's write down the sequence in a clearer way to identify the pattern:
6, (6 + 3 = 9), (9 + 131/2 = 19), (19 + 2 = 21), (21 + 3 = 24), (24 + 131/2 = 34), (34 + 2 = 36), ...
From this, we can observe that the terms can be generated by the following pattern:
First term: 6
Second term: First term + 3
Third term: Second term + 131/2
Fourth term: Third term + 2
Fifth term: Fourth term + 3
Sixth term: Fifth term + 131/2
Seventh term: Sixth term + 2
Using this pattern, we can write down the terms of the sequence up to the 13th term:
Term 1: 6
Term 2: 6 + 3 = 9
Term 3: 9 + 131/2 = 19
Term 4: 19 + 2 = 21
Term 5: 21 + 3 = 24
Term 6: 24 + 131/2 = 34
Term 7: 34 + 2 = 36
Term 8: 36 + 3 = 39
Term 9: 39 + 131/2 = 49
Term 10: 49 + 2 = 51
Term 11: 51 + 3 = 54
Term 12: 54 + 131/2 = 64
Term 13: 64 + 2 = 66
Now, we have all the terms up to the 13th term. To find the sum of these terms, we can simply add them up:
Sum of the 13 terms = (6 + 9 + 19 + 21 + 24 + 34 + 36 + 39 + 49 + 51 + 54 + 64 + 66)
Evaluating this equation gives us the final answer:
Sum of the 13 terms = 500